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Option Pricing with Fractional Stochastic Volatility and Discontinuous Payoff Function of Polynomial Growth

Author

Listed:
  • Viktor Bezborodov

    (The University of Verona)

  • Luca Persio

    (The University of Verona)

  • Yuliya Mishura

    (Taras Shevchenko National University of Kyiv)

Abstract

We consider the pricing problem related to payoffs of polynomial growth that can have discontinuities of the 1st kind. The asset price dynamic is modeled within the Black-Scholes framework characterized by a stochastic volatility term driven by a fractional Ornstein-Uhlenbeck process. In order to solve the aforementioned problem, we consider three approaches. The first one consists in a suitable transformation of the initial value of the asset price, in order to eliminate possible discontinuities. Then we discretize both the Wiener process and the fractional Brownian motion and estimate the rate of convergence of the related discretized price to its real value whose closed-form analytical expression is usually difficult to obtain. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Here we derive a presentation for the option price which involves only an integral functional depending on the fBm trajectory, and then discretize the fBm and estimate the rate of convergence of the associated numerical scheme. In both cases the rate of convergence is the same and equals n−rH, where n is the partition size, H is the Hurst index of the fBm, and r is the Hölder exponent of the volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus and providing the option price in terms of the associated probability density.

Suggested Citation

  • Viktor Bezborodov & Luca Persio & Yuliya Mishura, 2019. "Option Pricing with Fractional Stochastic Volatility and Discontinuous Payoff Function of Polynomial Growth," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 331-366, March.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:1:d:10.1007_s11009-018-9650-3
    DOI: 10.1007/s11009-018-9650-3
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    References listed on IDEAS

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    1. Elisa Nicolato & Emmanouil Venardos, 2003. "Option Pricing in Stochastic Volatility Models of the Ornstein‐Uhlenbeck type," Mathematical Finance, Wiley Blackwell, vol. 13(4), pages 445-466, October.
    2. Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Center for Research in Economics and Statistics.
    3. Ole E. Barndorff‐Nielsen & Neil Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 253-280, May.
    4. Carrasco, Marine & Chen, Xiaohong, 2002. "Mixing And Moment Properties Of Various Garch And Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 18(1), pages 17-39, February.
    5. Bollerslev, Tim & Ole Mikkelsen, Hans, 1996. "Modeling and pricing long memory in stock market volatility," Journal of Econometrics, Elsevier, vol. 73(1), pages 151-184, July.
    6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    7. Alexandra Chronopoulou & Frederi Viens, 2012. "Estimation and pricing under long-memory stochastic volatility," Annals of Finance, Springer, vol. 8(2), pages 379-403, May.
    8. Hideharu Funahashi & Masaaki Kijima, 2017. "Does the Hurst index matter for option prices under fractional volatility?," Annals of Finance, Springer, vol. 13(1), pages 55-74, February.
    9. F. Comte & L. Coutin & E. Renault, 2012. "Affine fractional stochastic volatility models," Annals of Finance, Springer, vol. 8(2), pages 337-378, May.
    10. Sergii Kuchuk-Iatsenko & Yuliya Mishura, 2015. "Pricing the European call option in the model with stochastic volatility driven by Ornstein--Uhlenbeck process. Exact formulas," Papers 1510.01848, arXiv.org.
    11. Rainer Schöbel & Jianwei Zhu, 1999. "Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension," Review of Finance, European Finance Association, vol. 3(1), pages 23-46.
    12. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    13. Jan Pospíšil & Tomáš Sobotka, 2016. "Market calibration under a long memory stochastic volatility model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 23(5), pages 323-343, September.
    14. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    15. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    Cited by:

    1. Yuliya Mishura & Kostiantyn Ralchenko & Sergiy Shklyar, 2020. "General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory," Risks, MDPI, vol. 8(1), pages 1-29, January.
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    4. Qi Zhao & Alexandra Chronopoulou, 2023. "Delta-hedging in fractional volatility models," Annals of Finance, Springer, vol. 19(1), pages 119-140, March.

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